Free Republic
Browse · Search
General/Chat
Topics · Post Article

Skip to comments.

The Unravelling of the Real 3D Mandelbulb [image rich]
Skytopia.com ^ | November 16, 2009 | Daniel White

Posted on 11/16/2009 5:09:59 AM PST by Daffynition

TIMESTAMP 08/11/2009. The original Mandelbrot is an amazing object that has captured the public's imagination for 30 years with its cascading patterns and hypnotically colourful detail. It's known as a 'fractal' - a type of shape that yields (sometimes elaborate) detail forever, no matter how far you 'zoom' into it (think of the trunk of a tree sprouting branches, which in turn split off into smaller branches, which themselves yield twigs etc.).

It's found by following a relatively simple math formula. But in the end, it's still only 2D and flat - there's no depth, shadows, perspective, or light sourcing. What we have featured in this article is a potential 3D version of the same fractal. For the impatient, you can skip to the nice pics, but the below makes an interesting read (with a little math as well for the curious).







TOPICS: Science
KEYWORDS: fractals; mandelbrot; science

Background

It's been almost two years since we last wrote about the potential for a real 3D equivalent to the famous 2D Mandelbrot set. We're talking about a fractal which produces exquisite detail on all axes and zoom levels; one that doesn't simply produce the 'extruded' look of the various height-mapped images, or the 'whipped cream' swirls of the Quaternion approach.

The article received a degree of popularity, but in the end, the problem remained open, and the Mandelbrot refused to relinquish its deepest secret. At times, it felt tempting to give up the search, but the potential for the thing is too great - we are talking about the Holy Grail of fractals after all.




Opening Pandora's Box For the Second Time

O

ur story starts with a guy named Rudy Rucker, an American mathematician, computer scientist and science fiction author (and in fact one of the founders of the cyberpunk science-fiction movement). Around 20 years ago, along with other approaches, he first imagined the concept behind the potential 3D Mandelbulb (barring a small mistake in the formula, which nevertheless still can produce very interesting results - see later), and also wrote a short story about the 3D Mandelbrot in 1987 entitled "As Above, So Below" (also see his blog entry and notebook). Back then of course, the hardware was barely up to the task of rendering the 2D Mandelbrot, let alone the 3D version - which would require billions of calculations to see the results, making research in the area a painstaking process to say the least.

So the idea slumbered for 20 years until around 2007. I then independently pictured the same concept and published the formula for the first time in November 2007 at the fractalforums.com web site. The basic idea is that instead of rotating around a circle (complex multiplication), as in the normal 2D Mandelbrot, we rotate around phi and theta in 3 dimensional spherical coordinates (see here for details). In theory, this could theoretically produce our amazing 3D Mandelbrot. But here's the somewhat disappointing result of the formula (click any of the pictures for a larger view):


The first thing I tried was multiplying phi and theta by two, resulting in the shape you see above. It's nice, but not exactly what I'd call a 3D Mandelbrot (zooming in doesn't show true 3D fractal detail).

This one is the same as to the left, except offsets have been added to the multiplication bit (0.5*pi to theta and 1*pi to phi), to make it appear almost 3D Mandelbrot-esque. Also see Thomas Ludwig's globally illuminated render, and this one from Krzysztof Marczak.


Same as the first, except this time we try only multiplying angle phi by two, but not theta.



Although the second one looks somewhat impressive, and has the appearance of a 3D Mandelbulb very roughly, we would expect the real deal to have a level of detail far exceeding it. Perhaps we should expect an 'apple core' shape with spheres surrounding the perimeter, and further spheres surrounding those, similar to the way that circles surround circles in the 2D Mandelbrot.

Zooming in reveals some interesting detail, but nothing truly fantastic. The best shot I could find was this view from the YZ plane (found just before this article was published actually):



Full size shown here. For other 'hot spots', try here, and this one from the inside.




Created by Dr. Kazushi Ahara and Dr. Yoshiaki. This looks great, but zooming in will not reveal the variety of style that the Mandelbrot has.

I went to great lengths to explore the concept, including the utilization of various spherical coordinate systems and adjusting the rotation of each point's 'orbit' after every half-turn of phi or theta. But it didn't work. Something was missing. I scoured everywhere to find signs of the 3D beast, but nothing turned up.

On October 13th, 2006, Marco Vernaglione put out the question and challenge to the world with this memorable document.

Pretty 3D fractals were everywhere, but nothing quite as organic and rich as the original 2D Mandelbrot. The closest turned out to be Dr. Kazushi Ahara and Dr. Yoshiaki's excellent Quasi-fuchsian fractal (see right), but it turned out that even that doesn't have the variety of the Mandelbrot after zooming in.



Is this merely a fool's quest?

Some said it couldn't be done - that there wasn't a true analogue to a complex field in three dimensions (which is true), and so there could be no 3D Mandelbulb. But does the essence of the 2D Mandelbrot purely rely on this complex field, or is there something else more fundamental to its form? Eventually, I also started to think that this was turning out to be a Loch Ness hunt. But there was still something at the back of my mind saying if this detail can be found by (essentially) going round and across a circle for the standard 2D Mandelbrot, why can't the same thing be done for a sphere to make a 3D version?

Our story continues with mathematician - Paul Nylander. His idea was to adjust the squaring part of the formula to a higher power, as is sometimes done with the 2D Mandelbrot to produce snowflake type results. Surely this can't work? After all, we'd expect to find sumptuous detail in the standard power 2 (square or quadratic) form, and if it's not really there, then why should higher powers work?

But maths can behave in odd ways, and intuition plays tricks on you sometimes. This is what he found (also see forum thread, and the full size pic at the 'Hypercomplex Fractals' page of his site):


Mandelbulb (order 8)


Okay... now this is starting to look interesting. We're already starting to see buds growing on buds. Could.... this... object still hold any fractal detail if we zoomed in far enough?! More of Paul's work can be found here.

Then something amazing happened.

Another fractal explorer, computer programmer David Makin was the first to render some sneak preview zooms of the above object, and this is what he found:

WOW! Okay, now we're talking. These are deep zoom levels (the first being over 1000x), but fractal details remain abundant in all three dimensions! The buds are growing smaller buds, and at least to the picture on right, there seems to be a great amount of variety too. We're seeing 'branches' with large buds growing around the branch in at least four directions. These in turn contain smaller buds, which themselves contain yet further tiny buds.

Remember, these pictures are not created from an iterated function system (IFS), but from a purely simple Mandelbrot-esque function!

Even the picture on the left is interesting, and is reminiscent of the Romanesco broccoli vegetable. But glance at the top right of the left picture - there also seems to be a leaf section in the shape of a seven sided star. Does this hint at a deeper variety in the object than we can possibly imagine? What the heck have we stumbled upon here?

Because of the lack of shadows, it's difficult for the renderings to give justice to the detail, but what we have here is a first look into a great unknown.

Eager to get a better look at this thing, I set about trying to find software to render it, preferably with full shadowing and even global illumination, and at least something that was fairly nippy. But it turns out that there are probably no 3D programs out there on the market that can render arbitrary functions, at least not with while loops and local variables (a prerequisite for anything Mandelbrot-esque!). So I set out to create my own specialized voxel-ish raytracer. Results could be slow (perhaps a week for 4000x4000 pixel renders!), but it'll be worth it right?

At first, I implemented 'fake' lighting based on the surface angle, and this produced a further glimpse into this incredible world (this one again from the power 8 version of the Mandelbulb):



Resulting renders

But it wasn't until I incorporated proper shadowing that the subtleties of this incredible object became apparent. For the renders below and exploration afterwards, I'll be concentrating mostly on the 8th power, since that seems to be around the 'sweet spot' for overall detail and beauty.


"Mandelbrot Crust" You don't have to zoom in far before you get to see this.

"A Slice of Mandelbrot Gateau" This picture is a deeper zoom of the previous picture (see its far right, just below vertically center - this one's near there.)


"Cave of Lost Secrets" This ancient half mile high cave still exists (now underwater) from a planet several billions of light years away from Earth. It was built by a (now extinct) intelligent race of beings who also discovered the 3D Mandelbulb we are witnessing on this page. Inside the cave however, lies - amongst other technological and mathematical secrets - the last remaining scroll which contains the much deeper secret of the even more incredible real 3D Mandelbrot formula (giant structures of that were also built at a later stage, but were apparently destroyed for reasons unknown).


"Magic Broccoli" Mini-Bulbs from the set don't just sprout uniform smaller bulbs, but rather twist and shear to create surprisingly strange patterns.

"Mandelbulb Garden" Zooming in yet further to the left, we visit the Bulb's horticultural centre, and still, we see no sign of detail degradation.


[below] Zooming out again, we see some of the main support structures holding up the giant 40 mile wide Colosseum at the top right of the picture. Seriously, this universe has got to be quite messed up to be harbouring math secrets capable of this kind of Baroquian beauty.


"Christmas Coral Egg" Red, green and blue lighting combine to create all the other colours in this chrimbo themed fractal.

"Christmas Coral Reef"


"Honeycomb Heaven" It's easy to think of a bee hive when seeing this. Little did I know at the time about the honeycomb pattern below the hive.


"Mandel NightShade" Rumour has it that one sniff of this plant and you're turned to dust. A little more deadly than usual then.


"The Mandelbulb" Here's the whole thing, with some perspective this time. It should be amazing to fly over and zoom into it. In the meantime, let's see what happens when if peek *inside* the Bulb.....

"Mandelbulb Spine" The inside is just as amazing as the out, as this zoom shows. A high definition, high res poster is available here (see this preview showing the detail of the 7000x7000 render).


"Ice Cream From Neptune" What, Neptune* from our solar system? Neah, we're talking about a planet unique in all but the name, near the distant edge of the universe. They have advanced food making equipment, and an eye for detail, so they regularly consume attractive cornetto-esque dishes, sometimes in the shape you see above.
* Yes we renamed it. We mistakingly used the planet's old name that was in use until around 50,000 years ago. By another massive but convenient coincidence, they themselves renamed it for similar reasons to Professor Farnsworth's story from Futurama.

"Caramelized Hazelnut Swirl" Objects take on a completely different character inside the 3D Mandelbulb, as these surrounding pictures testify.


"Shell Life" More chaotic scenes appear too. Whenever fractals are involved though, it's never going to be completely random. (...well unless we're talking about Quaternion Julia fractals [runs and ducks for cover] ;) )


"Hell Just Froze Over" 5 minutes ago, possibly because someone found the definitely-really-true-and-we-mean-it-this-time-3D .... Mandelbrot.



And here is the beast itself (power 8 version). All of the above images come from this object below (giant 4500x4500 pixel version available here). Added 13/11/2009 That version is low quality, but if you want the full size (7500x7500), high definition, high resolution version, you can buy the print from here.



It's not just myself of course who's rendered stunning shots of the Bulb. The guys have over at Fractal Forums have also rendered mouth watering pictures and animations, some of which I'll show below. It's a real honour to present them on this page - please visit their websites to view more of their creations.




This jaw dropping image created by Krzysztof Marczak uses many iterations to achieve a more fragmented surface texture. View full size, and you'll notice countless 'satellites' around other bigger satellites.


"The Honeycomb": The fantastic material combined with the different colour ranges give a real sense of depth in this picture created by David Makin.


"Siebenfach": The unusual material and ornate rope-like detail evokes a more mysterious atmosphere in this stunning render from Thomas Ludwig. Full resolution available from here.


Gotta love the luminous sorbet style texture of the quadratic version of the Mandelbulb, created by Paul Nylander. See his Hypercomplex Fractals page for a bigger view.


"Asteroid National Park" - This degree 4 behemoth would have certainly made an interesting replacement for the asteroids used in "Armageddon" and "Deep Impact". Excellent render by David Makin; view full res to see it in all its glory.


Created by Garth Thornton, a special variant of the Julia formula is used in combination here to create this amazing fossilized design. See the thread here for further interesting variations.



Keep an eye out for the latest news and renders on this page, and over in the 3D Mandelbrot thread over at Fractal Forums. As of 05/11/2009, the pictures here in this article are only a preliminary look into the thing. The best stuff is surely yet to come!



Page 2 - Further exploration of the 3D Mandelbulb






Most pictures on this page are copyright Daniel White 2009 onwards, apart from
where attribution is made - where they are copyright of their respective owners.



1 posted on 11/16/2009 5:09:59 AM PST by Daffynition
[ Post Reply | Private Reply | View Replies]

To: Daffynition

I gotta tell ya, I don’t understand any of this.

But it sure makes some pretty pictures!


2 posted on 11/16/2009 5:17:54 AM PST by Roccus (My anger is manufactured.......................................in the WHITE HOUSE and CONGRESS!!)
[ Post Reply | Private Reply | To 1 | View Replies]

To: Daffynition

Kinda neat to look at but I’ve never understood the point.


3 posted on 11/16/2009 5:18:30 AM PST by cripplecreek (Seniors, the new shovel ready project under socialized medicine.)
[ Post Reply | Private Reply | To 1 | View Replies]

To: Daffynition

Even the author isn’t sure that this is real 3D Mandelbrot..but it sure looks good!


4 posted on 11/16/2009 5:22:58 AM PST by pvoce ('Good' sense and 'Common' sense are two entirely different concepts.)
[ Post Reply | Private Reply | To 1 | View Replies]

To: Roccus

Well this is 3D at its very best. The hardest work of a computer that puts a 3D picture or even a 3D video together is the “rendering process”. It takes a lot of memory.


5 posted on 11/16/2009 5:24:46 AM PST by Biggirl (Political Correctness KILLS! =^..^==^..^==^..^==^..^==^..^=)
[ Post Reply | Private Reply | To 2 | View Replies]

To: Daffynition

THANK YOU FOR THIS POST (and my newest screensaver images)


6 posted on 11/16/2009 5:26:45 AM PST by Mr. K (Deathly afraid one of my typos becomes a freeper catchphrase...I'm series!)
[ Post Reply | Private Reply | To 1 | View Replies]

To: Daffynition

Reminds me of using an electrion microscope to image a diatom when I was in college.


7 posted on 11/16/2009 5:29:45 AM PST by Red_Devil 232 (VietVet - USMC All Ready On The Right? All Ready On The Left? All Ready On The Firing Line!)
[ Post Reply | Private Reply | To 1 | View Replies]

To: cripplecreek
Kinda neat to look at but I’ve never understood the point.

Yep...I would certainly like to understand if this has any practical uses other than art. Frankly, I'm just ignorant on the subject and would like to learn more.

8 posted on 11/16/2009 5:32:17 AM PST by Wyatt's Torch (I can explain it to you. I can't understand it for you.)
[ Post Reply | Private Reply | To 3 | View Replies]

To: Daffynition

.."that ah...terd one of da left looks hauntingly familure"

9 posted on 11/16/2009 5:52:35 AM PST by Doogle (USAF.68-73..8th TFW Ubon Thailand..never store a threat you should have eliminated))
[ Post Reply | Private Reply | To 1 | View Replies]

To: Daffynition
This calls to mind Guadi's Sagrada Familia cathedral in Barcelona


10 posted on 11/16/2009 5:53:15 AM PST by Notwithstanding (Wer glaubt ist nie allein. Who believes is never alone.)
[ Post Reply | Private Reply | To 1 | View Replies]

To: Daffynition

Reminds me of the album covers for “Yes” that Roger Dean created.


11 posted on 11/16/2009 5:56:30 AM PST by manic4organic (Obama shot hoops, America lost troops.)
[ Post Reply | Private Reply | To 1 | View Replies]

To: cripplecreek

The point is that simple functions iterated endlessly can produce seemingly infinitely complex structures.

In other words, the foundation of endlessly complexity is beautiful simplicity.


12 posted on 11/16/2009 5:57:55 AM PST by DManA
[ Post Reply | Private Reply | To 3 | View Replies]

To: manic4organic

13 posted on 11/16/2009 5:57:55 AM PST by Wyatt's Torch (I can explain it to you. I can't understand it for you.)
[ Post Reply | Private Reply | To 11 | View Replies]

To: cripplecreek

Let me try that again.

The point is that simple functions iterated endlessly can produce seemingly infinitely complex structures.

In other words, hidden within endlessly complexity is beautiful simplicity.


14 posted on 11/16/2009 5:58:31 AM PST by DManA
[ Post Reply | Private Reply | To 3 | View Replies]

To: Daffynition; MotleyGirl70; Cagey; earlJam; F15Eagle

Mandelbulb! Mandelbulb! Mandelbulb!

(Sorry for the Seinfeld hijack. Couldn’t help myself. We now return you to your original and really cool thread). :-)


15 posted on 11/16/2009 6:00:18 AM PST by Larry Lucido
[ Post Reply | Private Reply | To 1 | View Replies]

To: DManA

OK

I like green but blue is cool too.

LOL


16 posted on 11/16/2009 6:00:34 AM PST by cripplecreek (Seniors, the new shovel ready project under socialized medicine.)
[ Post Reply | Private Reply | To 14 | View Replies]

To: Daffynition

Bookmark for later viewing.


17 posted on 11/16/2009 6:02:35 AM PST by Reaganesque ("And thou shalt do it with all humility, trusting in me, reviling not against revilers.")
[ Post Reply | Private Reply | To 1 | View Replies]

To: Larry Lucido; Cagey; MotleyGirl70
Sorry for the Seinfeld hijack.

Not that there's anything wrong with that....

18 posted on 11/16/2009 6:10:45 AM PST by earlJam
[ Post Reply | Private Reply | To 15 | View Replies]

To: Larry Lucido; Daffynition; earlJam; MotleyGirl70

Vile Weed!

19 posted on 11/16/2009 6:21:24 AM PST by Cagey
[ Post Reply | Private Reply | To 15 | View Replies]

To: Cagey

MR. PITT: Elaine, did I hear... (sees Kramer) Oh, this is very odd.

KRAMER: (looking at picture) Yeah, it’s 3-D art. Computers generate ‘em. BIG computers.

MR. PITT: Yes, I’ve heard about these. How do they work?

KRAMER: Well, you blur your eyes like you’re starin’ straight through the picture. And you keep your eyes unfocused. And then... (Kramer and Pitt stare at picture) Oh, oh, oh, YEAH!

MR. PITT: I don’t see it.

KRAMER: Yeah, it’s a spaceship, surrounded by planets, asteroids...

MR. PITT: I still don’t see it.

ELAINE: Okay, Kramer, that’s enough. Mr. Pitt has got work to do.

KRAMER: Ya’ ever dream in 3-D? It’s like the boogeyman is comin’ right at you.

MR. PITT: A spaceship, where?

KRAMER: (pointing) Right in here. Just keep your eyes unfocused. (convulses in pain) Waahh! Oh, mama!


20 posted on 11/16/2009 6:32:20 AM PST by Larry Lucido
[ Post Reply | Private Reply | To 19 | View Replies]

To: Larry Lucido

Hahahahahaha!!


21 posted on 11/16/2009 6:33:27 AM PST by Cagey
[ Post Reply | Private Reply | To 20 | View Replies]

To: Daffynition

bump


22 posted on 11/16/2009 6:33:37 AM PST by dangerdoc
[ Post Reply | Private Reply | To 1 | View Replies]

To: 21stCenturion

...


23 posted on 11/16/2009 6:52:36 AM PST by 21stCenturion ("It's the Judges, Stupid !")
[ Post Reply | Private Reply | To 1 | View Replies]

To: Daffynition
Let Them Eat Pixels


24 posted on 11/16/2009 10:17:31 AM PST by JoeProBono (A closed mouth gathers no feet)
[ Post Reply | Private Reply | To 1 | View Replies]

To: Cagey
“I do not like broccoli. And I haven't liked it since I was a little kid and my mother made me eat it. And I'm President of the United States and I'm not going to eat any more broccoli.”


25 posted on 11/16/2009 1:03:43 PM PST by Daffynition (What's all this about hellfire and Dalmatians?)
[ Post Reply | Private Reply | To 19 | View Replies]

To: JoeProBono

26 posted on 11/16/2009 1:05:12 PM PST by Daffynition (What's all this about hellfire and Dalmatians?)
[ Post Reply | Private Reply | To 24 | View Replies]

To: Doogle

LOL ........you’re B-A-D!

LOL


27 posted on 11/16/2009 4:48:52 PM PST by Daffynition (What's all this about hellfire and Dalmatians?)
[ Post Reply | Private Reply | To 9 | View Replies]

To: Larry Lucido

PERFECT! HAHAHA!


28 posted on 11/16/2009 4:49:53 PM PST by Daffynition (What's all this about hellfire and Dalmatians?)
[ Post Reply | Private Reply | To 15 | View Replies]

To: Daffynition; JoeProBono; BGHater; decimon; Ernest_at_the_Beach; blam; neverdem

thanks Daffynition.

http://www.digitalblasphemy.com/

http://www.mikebonnell.com/


29 posted on 11/16/2009 6:56:15 PM PST by SunkenCiv (https://secure.freerepublic.com/donate/__Since Jan 3, 2004__Profile updated Monday, January 12, 2009)
[ Post Reply | Private Reply | To 1 | View Replies]

To: SunkenCiv

Which Mandel brot is wurst?


30 posted on 11/17/2009 6:34:25 AM PST by decimon
[ Post Reply | Private Reply | To 29 | View Replies]

To: Daffynition

BTTT for mandelbrots!


31 posted on 11/29/2009 5:16:05 PM PST by texas booster (Join FreeRepublic's Folding@Home team (Team # 36120) Cure Alzheimer's!)
[ Post Reply | Private Reply | To 1 | View Replies]

Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.

Free Republic
Browse · Search
General/Chat
Topics · Post Article

FreeRepublic, LLC, PO BOX 9771, FRESNO, CA 93794
FreeRepublic.com is powered by software copyright 2000-2008 John Robinson